BSTJ 16: 4. October 1937: The Dielectric Properties of Insulating Materials. (Murphy, E.J.; Morgan, S.O.)

The Dielectric Properties of Insulating Materials

By E. J. MURPHY and S. O. MORGAN

This paper gives a qualitative account of the way in which
dielectric constant and absorption data have been interpreted in
terms of the physical and chemical structure of materials. The
dielectric behavior of materials is determined by the nature of the
polarizations which an impressed field induces in them. The
various types of polarization which have been demonstrated to exist
are listed, together with an outline of their characteristics.

I. Outline of the Physico-Chemical Interpretation
of the Dielectric Constant

THE development of dielectric theory in recent years has been
along such specialized lines that there is need of some correlation
between the newer and the older theories of dielectric behavior to
keep clear what is common to both, though sometimes expressed in
different terms. The purpose of the present paper is to outline in
qualitative terms the way in which the dielectric constant varies with
frequency and temperature and to indicate the type of information
regarding the structure of materials which can be obtained from the
study of the dielectric constant.

The important dielectric properties include dielectric constant (or
specific inductive capacity), dielectric loss, loss factor, power factor,
a.c. conductivity, d.c. conductivity, electrical breakdown strength and
other equivalent or similar properties. The term dielectric behavior
usually refers to the variation of these properties with frequency,
temperature, voltage, and composition.

In discussing the dielectric properties and behavior of insulating
materials it will be necessary to use some kind of model to represent
the dielectric. The success of wave-mechanics in explaining why
some materials are conductors and others dielectrics suggests that it
might be desirable to use a quantum-mechanical model even in a
general outline of the characteristics of dielectrics, but for the aspects
of the theory of dielectric behavior with which we are immediately
concerned here the behavior predicted is essentially the same as that
derived on the basis of classical mechanics. However, in the course
of the description of the frequency-dependence of dielectric constant
we shall have occasion to make a comparison between the dispersion

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494 BELL SYSTEM TECHNICAL JOURNAL

and absorption curves for light and those for electromagnetic dis-
turbances in the electrical (i.e., radio and power) range of frequencies.
The difficulty is then met that the quantum-mechanical model is
the customary medium of description of the absorption of light.
But, since the references to optical properties will be only incidental
and for comparative purposes, there is little to be lost, even in this
domain in which quantum-mechanical concepts are the familiar
medium of description, in using the pre-quantum theory concepts of
dispersion and absorption processes. Thus a model operating on the
basis of classical mechanics and the older conceptions of atomic struc-
ture will be sufficient for our present purposes.

On the wave-mechanical theory of the structure of matter a di-
electric is a material which is so constructed that the lower bands of
allowed energy levels are completely full at the absolute zero of temper-
ature (on the Exclusion Principle) and at the same time isolated from
higher unoccupied bands by a large zone of forbidden energy levels. 1
Thus conduction in the lower, fully occupied bands is impossible
because there are no unoccupied energy levels to take care of the
additional energy which would be acquired by the electrons from the
applied field, while the zone of forbidden energy levels is so wide that
there is only a negligible probability that an electron in the lower band
of allowed levels will acquire enough energy to make the transition to
the unoccupied upper band where it could take part in conduction.
The bound electrons in a completely filled and isolated band of allowed
levels can, however, interact with the applied electric field by means of
the slight modifications which the applied field makes in the potential
structure of the material and hence in the allowed levels.

On the other hand in the older theory of the structure of matter the
essential condition which makes a material a dielectric is that the
electrons and other charged particles of which it is composed are held
in equilibrium positions by constitutive forces characteristic of the
structure of the material. When an electric field is applied these
charges are displaced, but revert to their original equilibrium positions
when the field is removed. In this account of the behavior of di-
electrics this model will be sufficient, but no essential change in the
relationships which will be discussed here would result if a translation
were made to a model based upon quantum-mechanics.

When an electric field is impressed upon a dielectric the positive

and negative charges in its atoms and molecules are displaced in

opposite directions. The dielectric is then said to be in a polarized

1 Cf., for example, Gurney, "Elementary Quantum Mechanics," Cambridge
(1934); Herzfeld, "The Present Theory of Electrical Conduction," Electrical Engi-
neering, April 1934.

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 495

condition, and since the motion of charges of opposite sign in opposite
directions constitutes an electric current there is what is called a
polarization current or charging current flowing while the polarized
condition is being formed.

For the case of a static impressed field a charging current flows in
the dielectric only for a certain time after application of the field, the
time required for the dielectric to reach a fully polarized condition.
If the material is not an ideal dielectric, but contains some free ions,
the current due to a static impressed field does not fall to zero but to
a constant value determined by the conductivity due to free ions.
More important than the static is the alternating current case, where
the potential is continually varying and where, consequently, there
must be a continuously varying current.

The dielectric behavior of different materials under different con-
ditions is reflected in the characteristics of the charging or polariza-
tion currents, but since polarization currents depend upon the applied
voltage and the dimensions of condensers it is inconvenient to use
them directly for the specification of the properties of materials.
Eliminating the dependence upon voltage by dividing the charge by
the voltage, we have the capacity (C = Q/V); and the dependence
upon dimensions may be eliminated by using the dielectric constant,
defined as e = C/C , where C is the capacity of the condenser when the
dielectric material is between its plates and Co is the capacity of the
same arrangement of plates in a vacuum. The dielectric constant
is then a property of the dielectric material itself.

The term "dielectric polarization" is used to refer to the polarized
condition created in a dielectric by an applied field of either constant
or varying intensity. The polarizability is one of the quantitative
measures of the dielectric polarization; it is defined as the electric
moment per unit volume induced by an applied field of unit effective
intensity. Another quantitative measure of the dielectric polarization
is the molar polarization; this is a quantity which is a measure of the
polarizability of the individual molecule, whatever the state of ag-
gregation of the material.

The concept of polarizability is as fundamental to, and plays about
the same role in, the theory of dielectric behavior as does the concept
of free ions in the theory of electrolytic conduction. Just as the con-
ductivity of a material is a measure of the product of the number of
ions per unit cube and their average velocity in the direction of a unit
applied field, so the polarizability is a measure of the number of bound
charged particles per unit cube and their average displacement in the
direction of the applied field.

496 BELL SYSTEM TECHNICAL JOURNAL

In the early investigations of dielectrics two distinct types of charg-
ing current were recognized, the one in which the charging or dis-
charging of a condenser occurred practically instantaneously and the
other in which a definite and easily observable time was required. A
charge accumulating in a condenser in an unmeasurably short time
was variously referred to as the instantaneous charge or geometric
charge or the elastic displacement. The current by which this charge
is formed was called the instantaneous or geometric charging current,
and similarly the terms instantaneous dielectric constant or geometric
dielectric constant were used to describe the property of the medium
giving rise to the effect between the condenser plates. An even wider
variety of names has been used for the part of the charge which formed
or disappeared more slowly. Among these are residual charge,
reversible absorption, inelastic displacement, viscous displacement
and anomalous displacement. The modern theory still recognizes
these two distinct types of condenser charges and charging currents
but the simple descriptive designations rapidly-forming or instantaneous
polarizations and slowly-forming or absorptive polarizations will be
adopted here, as they seem sufficient and to be preferred to terms
which have more specialized connotations as to the mechanism upon
which the behavior depends. The properties of these two types of
charging currents and the dielectric polarizations corresponding to
them appear prominently in the theories of dielectric behavior.

The total polarizability of the dielectric is the sum of contributions
due to all of the different types of displacement of charge produced in
the material by the applied field. Constitutive forces characteristic
of the material determine both the magnitude of the polarizability and
the time required for it to form or disappear. The quantitative
measure of the time required for a polarization to form or disappear is
called the relaxation-time. In the following a description will be given
of the physical processes involved in the formation of dielectric polari-
zations, indicating the effect of chemical and physical structure upon
the two quantities, magnitude and relaxation-time, which determine
many of the properties of dielectric polarizations of the slowly-forming
or absorptive type.

The magnitude of the polarizability k of a dielectric can be expressed
in terms of a directly measurable quantity, the dielectric constant e,
by the relation

4tt (e + 2)
It is sometimes convenient to use the polarizability and the dielectric

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 497

constant interchangeably in the qualitative discussion of the magnitude
of the dielectric polarization. In dealing with alternating currents
the fact that polarizations of the absorptive type require a time to
form which is often of the same order of magnitude as, or greater than,
the period of the alternations, results in the polarization not
being able to form completely before the direction of the field is
reversed. This causes the magnitude of the dielectric polarization

Fig. 1 — Schematic diagram of variation of dielectric constant and dielectric
absorption with frequency for a material having electronic, atomic, dipole and
interfacial polarizations.

and dielectric constant to decrease as the frequency of the applied
field increases. An example of this variation of the dielectric constant
with frequency is shown in the radio and power frequency section of
the curve plotted in Fig. 1. It is often convenient to refer to the mid-
point of the decreasing dielectric constant-frequency curve as the
relaxation-frequency; this frequency f m is very simply related to the
relaxation- time t, for the theory of these effects shows that/ m = 1/2 tt.

498 BELL SYSTEM TECHNICAL JOURNAL

Various types of polarization can be induced in dielectrics: There
should be an electronic polarization due to the displacement of electrons
with respect to the positive nuclei within the atom; an atomic polari-
zation due to the displacement of atoms with respect to each other in
the molecule and in certain ionic crystals, such as rock salt, to the
displacement of the lattice ions of one sign with respect to those of the
opposite sign; dipole polarizations due to the effect of the applied
field on the orientations of molecules with permanent dipole moments;
and finally interfacial (or ionic) polarizations caused by the accumula-
tion of free ions at the interfaces between materials having different
conductivities and dielectric constants.

Electronic Polarizations

A classification of dielectric polarizations into rapidly-forming or
instantaneous polarizations and slowly-forming or absorptive polariza-
tions has been made. Instantaneous polarizations may be thought of
as polarizations which can form completely in times less than say 10~ 10
seconds, that is, at frequencies greater than 10 10 cycles per second or
wave-lengths of less than 1 centimeter, and so beyond the range of
conventional dielectric constant measurements. The electronic polari-
zations are due to the displacement of charges within the atoms, and
are the most important of the instantaneous polarizations. The
polarizability per unit volume due to electronic polarizations may be
considered to be a quantity which is proportional to the number of
bound electrons in a unit volume and inversely proportional to the
forces binding them to the nuclei of the atoms.

The effect of number of electrons and binding force is illustrated by
a comparison of the values for the polarizability per unit volume of
different gases; for the number of molecules per unit volume is inde-
pendent of the composition of the gas. Thus, although a c.c. of
hydrogen with two electrons per molecule has the same number of
electrons as a c.c. of helium, which is an atomic gas with two electrons
per atom, the quantity e— 1, that is the amount by which the di-
electric constant is greater than that of a vacuum, is nearly four times
as large for hydrogen as for helium. This shows that in hydrogen the
electrons are in effect less tightly bound to the nucleus than in helium,
resulting in a larger induced polarization. Nitrogen has a. larger
dielectric constant than either hydrogen or helium because it has 14
electrons per molecule. Some of these are tightly bound as in helium
and some are more loosely bound as in hydrogen.

The dielectric constant of liquid nitrogen is 1.43, which is much
higher than the value 1.000600 for the gas. This is due to the fact

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 499

that the number of molecules, and consequently of bound charges,
per unit volume is much greater in the liquid than in the gas. How-
ever, the molar polarization, a quantity which is corrected for varia-
tions in density, is the same for liquid as for gaseous nitrogen.

The time required for the applied field to displace the electrons
within an atom to new positions with respect to their nuclei is so short
that there is no observable effect of time or frequency upon the value
of the dielectric constant until frequencies corresponding to absorption
lines in the visible or ultra-violet spectrum are reached. For con-
venience in this discussion the frequency range which includes the
infra-red, visible and ultra-violet spectrum will be called the optical
frequency range while that which includes radio, audio and power
frequencies will be called the electrical frequency range. For all fre-
quencies in the electrical range the electronic polarization is indepen-
dent of frequency and for a given material contributes a fixed amount
to the dielectric constant, but at the frequencies in the optical range
corresponding to the absorption lines in the spectrum of the material,
the dielectric constant, or better the refractive index, changes rapidly
with frequency, and absorption appears. (The justification for using
refractive index n and dielectric constant e interchangeably for the
qualitative discussion of the properties of dielectric polarizations fol-
lows from the relation, e = n 2 , which is known as Maxwell's rule.
This is a general relationship based upon electromagnetic theory and
is applicable whenever e and n are measured at the same frequency
no matter how high or low it may be.)

The electronic polarization of a molecule may be regarded as an
additive property of the atoms or of the atomic bonds in the molecule,
and may be calculated for any dielectric of known composition with
sufficient accuracy for most purposes. Within any one chemical class of
compounds such as, for example, the saturated hydrocarbons or their
simple derivatives, in which all of the bonds are C — H, C — C or C — X,
the calculated values agree with the measured to within a few per
cent. For other classes of compounds — for example, benzene, in
which there are both single and double bonds such calculations must be
corrected for the fact that some of the valence electrons have their
binding forces and hence their polarizabilities altered in the double
bond as compared to the single bond. Such values of electronic
polarization, usually called atomic refractions, have been determined
for all of the different types of bonds from the vast amount of experi-
mental study of refractive indices of organic and inorganic compounds.

In some materials the electronic polarization is the only one of
importance. For example, in benzene the dielectric constant is the

500 BELL SYSTEM TECHNICAL JOURNAL

same at all frequencies in the electrical range and is equal to the square
of the optical refractive index. This must mean that the only polari-
zable elements of consequence in CeH 6 are electrons which are capable
of polarizing as readily in the visible spectrum, where the refractive
index is measured, as at lower frequencies where dielectric constant is
measured. The refractive index in the visible spectrum provides the
means of determining the magnitude of electronic polarizations, for
other types of polarization are usually of negligible magnitude when
the frequency of the impressed field lies in the visible spectrum. For
materials having only electronic polarizations the dielectric properties
are very simply dependent upon the chemical composition and the
temperature, and are independent of frequency in the electrical
frequency range. In many materials, however, there are also other
polarizations which can form at low frequencies but not at high ; these
are characterized by more complex dielectric behavior.

Atomic Polarizations

Included among the polarizations which may be described as in-
stantaneous by comparison with the order of magnitude of the periods
of alternation of the applied field in the electrical frequency range are
those arising from the displacement of the ions in an ionic crystal
lattice (such as rock salt) or of atoms in a molecule or molecular lattice.
In some few materials, for example the alkali halides, sufficient study
has been made of the infra-red refractive index to provide data on the
atomic polarizations, but for most substances little is known about
them. What is known has in part been inferred from infra-red absorp-
tion spectra and in part from the infra-red vibrations revealed by
studies of the Raman effect.

Atomic polarizations are distinguished from electronic polarizations
by being the part of the polarization of a molecule which can be at-
tributed to the relative motion of the atoms of which it is composed.
The atomic polarizations may be attributed to the perturbation by the
applied field of the vibrations of atoms and ions having their character-
istic or resonance frequencies in the infra-red. Atomic polarizations
may be large for substances such as the alkali halides and other in-
organic materials, but are usually negligible for organic materials.
The exact value of the time required for the formation of atomic
polarizations is unimportant in the electric range of frequencies with
which we are primarily concerned. The essential thing is that atomic
polarizations begin to contribute to e(or «*) at frequencies below
approximately 10 14 seconds — that is, in the near infra-red and that
below about 10 10 cycles per second, where the optical and electrical

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 501

frequency ranges merge, atomic polarizations contribute a constant
amount to e(or n 2 ) for a given material. The atomic polarization is
determined as the difference between the polarization which is meas-
ured at some low infra-red or high electric frequency and the electronic
polarization as determined from refractive index measurements in the
visible spectrum.

The electronic and atomic polarizations are considered to comprise
all of the so-called instantaneous polarizations; that is, the polariza-
tions which form completely in a time which is very short as compared
with the order of magnitude of the periods of applied fields in the
electrical range of frequencies.

The Debye Orientational Polarization
The remaining types of polarization are of the "absorptive" kind,
characterized by relaxation-times corresponding to "relaxation-
frequencies" in the electrical range of frequencies. These polariza-
tions include the important type which is due to the effect of the applied
field on the orientation of molecules with permanent electric moments,
the theory of which was developed by Debye. Among the other
possible polarizations of the absorptive type are those due to inter-
facial effects or to ions which are bound in various ways.

Debye, 2 in 1912, suggested that the high dielectric constant of water,
alcohol and similar liquids was due to the existence of permanent
dipoles in the molecules of these substances. The theory which Debye
based upon this postulate opened up a new field for experimental
investigation by providing a molecular mechanism to explain dielectric
behavior which fitted into and served to confirm the widely held
views of chemical structure. Debye postulated that the molecules
of all substances except those in which the charges are symmetrically
located possess a permanent electric moment which is characteristic
of the molecule. In a liquid or gas these molecular dipoles are oriented
at random and therefore the magnitude of the polarization vector is
zero. When an electric field is applied, however, there is a tendency
for the molecules to align themselves with their dipole axes in the
direction of the applied field, or, put in another way, to spend more
of their time with their dipole axes in the direction of the field than
in the opposite direction. This dipole polarization is superimposed
upon the electronic and atomic polarizations which are also induced by
the field. The theory as developed by Debye accounts for the ob-
served difference between the temperature and frequency dependence
of the dipole polarizations and the instantaneous polarizations. While
the latter are present in all dielectrics, the dipole polarizations can
2 P. Debye, Phys. Zeit., 13, 97, (1912); Verh. d. D. phys. Ges., 15, 777 (1913).

502

BELL SYSTEM TECHNICAL JOURNAL

occur only in those made up of molecules which are electrically
asymmetrical.

Polar molecules (that is molecules with permanent electric moments)
are, by definition, those in which the centroid of the negative charges
does not coincide with the centroid of the positive charges, but falls
at some distance from it. All materials must be classed either as
polar or non-polar, the latter class including those which are elec-
trically symmetrical. Some simple examples of non-polar molecules

METHYL CHLORIDE
(CH3CI)

Fig. 2 — Methane and carbon tetrachloride are non-polar molecules each having
four equal vector moments whose sum is zero. Methyl chloride is polar because the
sum of the vector moments is not zero.

are H 2 , N 2 , 2 , CH 4 , CC1 4 and C 6 H 8 . In these molecules each C — H,
C — CI or other bond may be regarded as having a vector dipole mo-
ment of characteristic magnitude located in the bond. Where the
sum of these vector moments is zero the molecule will be non-polar.
Both CH 4 and CC1 4 meet this requirement but CH 3 C1 is polar because
the C — CI vector moment is considerably greater than the resultant of
the three C — H vectors. (See Fig. 2.) Polar molecules are the rule
and non-polar the exception.

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 503

In the discussion of dipole polarizations it has frequently been
pointed out that non-polar materials usually obey the general relation-
ship e = n 2 whereas for polar materials such as H 2 0, NH 8 and HC1
this rule is apparently not obeyed. Water, for example, has n 2 = 1.7
and e = 78. This apparent discrepancy arises because the refractive
index as measured in the visible spectrum is usually compared with
the dielectric constant as measured in the electric range of frequencies.
Non-polar materials usually have only electronic polarizations and
these can form both in the optical and in the electrical frequency
ranges, but the dipole polarizations can form and contribute to the
dielectric constant only in the electrical frequency range; this is the
most frequent source of the above mentioned discrepancy. The
general relationship e = n 2 should apply for any material at any fre-
quency provided e and n are measured at the same frequency. The
refractive index of water when measured with electric waves, 3 for
example, at a million cycles, is found to be slightly less than 9, the
square of which agrees very well with the observed value e = 78.
However, it does not always follow that when e > n 2 the molecules of
which the material is composed have permanent dipole moments, for
this condition can also result from the presence of any slowly-forming
or absorptive polarization or of a large atomic polarization. Experi-
mental investigations based upon the Debye theory have shown,
however, that in the case of water and many other familiar compounds
the orientation of dipole molecules actually accounts for the high
dielectric constant.

The Debye theory shows that the magnitude of the dipole polariza-
tion of a material is proportional to the square of the electric moment
of the molecule, which, as has been pointed out, may be regarded as
the vector sum of a number of constituent moments characteristic of
the individual atoms or radicals of which the molecule is composed,
or alternatively, of the bonds which bind these atoms into molecules
or more complex aggregates. The very great amount of experimental
study of the Debye theory has shown that the N0 2 and CN groups
have the largest group moments while CO, OH, NH 2 , CI, Br, I and
CH 3 have progressively smaller group moments. The value 34 for the
dielectric constant of nitrobenzene (C 6 H 5 N02), as against 5.5 for
chlorobenzene (C 6 H 5 C1), 2.8 for methyl benzene (C 6 H 5 CH 3 ) and 2.28
for benzene (C 6 H 6 ), which is non-polar, are evidence of the large
differences in the magnitudes of these group moments and the large
part that dipole moments can play in determining the dielectric
constant.

» Drude, "Physik des Aethers," Stuttgart (1894), p. 486.

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BELL SYSTEM TECHNICAL JOURNAL

Another point regarding molecular structure shown by such studies
is that it is not only the presence of polar groups in the molecule but
also their position which determines the electric moment of the mole-
cule. This is nicely illustrated by the dichlorobenzenes, of which
there are three isomers. As is shown in Fig. 3, ortho-dichlorobenzene,
having the two substituent groups in adjacent positions, is the most
asymmetrical of the three compounds, and consequently has the high-

PARA

Fig. 3 — Ortho dichlorobenzene being the more asymmetrical has a higher electric
moment than the meta isomer; the para isomer which is symmetrical has zero electric
moment.

est electric moment, n = 2.3. The meta compound has about the
same moment as monochlorobenzene, /z = 1.55. The para compound,
however, is symmetrical and has zero electric moment because the
CI atoms are substituted on opposite sides of the benzene ring so that
their vector moments cancel. These values of electric moment are
reflected in the values of dielectric constant which are respectively
10, 5.5 and 2.8 for the three isomeric dichlorobenzenes.

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 505

Dielectric studies of this kind have also shown, for example, that
H2O is not a symmetrical linear molecule, H — O — H, but rather a

/H

triangular structure 0<C . CO2 on the other hand, being non-polar,

Nh

is determined to be a linear molecule O = C = O. Thus, dielectric
measurements interpreted by the Debye theory have become estab-
lished as one of the standard means of studying molecular structure.

Since dipole polarizations depend upon the relative orientations of
molecules, rather than upon the displacement of charges within the
atom or molecule, the time required for a polarization of this type to
form depends upon the internal friction of the material. Debye
expressed the time of relaxation of dipole polarizations in terms of the
internal frictional force by the equation:

f Swrja 3

2kT 2kT '

where f is the internal friction coefficient, 77 is the coefficient of viscosity,
a the radius of the molecule and T the absolute temperature. 4 This
latter expression, because it depends on Stokes' law for a freely falling
body, is rigidly applicable only to gases or possibly to dilute solutions
of polar molecules in non-polar solvents in which the polar molecules
are far enough apart that they exert no appreciable influence on each
other.

Applying this equation to the calculation of the relaxation-time of
the orientational polarizations in water at room temperature we obtain
t = 10~ 10 seconds, assuming a molecular radius of 2 X 10~ 8 cm. and
taking the viscosity as 0.01 poises. 8 The relaxation-frequency corre-
sponding to this relaxation-time is about 1.6 X 10 9 cycles/sec, agreeing
with the results of experimental studies on water which show that
in the range of frequencies extending from 10 9 to 10 u cycles the
dielectric constant decreases from its high value to a value approxi-
mately equal to the square of the refractive index. Thus the drop in
dielectric constant occurs in the frequency range which corresponds to
the calculated value of the relaxation-time.

Similar experiments on dilute solutions of alcohols 6 in non-polar
solvents yield values of r of about 10~ 9 seconds. The shortest relax-
ation-times which dipole polarizations can have are probably not

* P. Debye, "Polar Molecules," Chem. Cat. Co., 1929, p. 85.

6 The viscosity of a liquid in poises is given by the force in dynes required to
maintain a relative tangential velocity of 1 cm. /sec. between two parallel planes in the
liquid each 1 cm. 2 in area and 1 cm. apart, the distance being measured normal to their
surfaces.

6 R. Goldammer, Phys. Zeit., 33, 361 (1932).

506 BELL SYSTEM TECHNICAL JOURNAL

much less than the order of 10 -u seconds, since in general either the
internal friction or the molecular radius of materials having polar
molecules will be greater than those of water, resulting in longer
relaxation-times. No long-time limit can be placed on the relaxation-
times which dipole polarizations may have, for they are limited only
by the values which the internal friction can assume. For materials,
such as glycerine, which tend to become very viscous at low tempera-
tures the time of relaxation of the dipoles may be a matter of minutes.
Studies of the dielectric constant of crystalline solids, to be discussed
in a later paper, show also that in some cases polar molecules are able
to rotate even in the crystalline state, where the ordinary coefficient
of viscosity has no meaning because the materials do not flow. In
connection with the dielectric properties we are concerned only with
the ability of the polar molecules to undergo rotational motion and
it is likely that in these solids, which constitute a special class, the
internal frictional force opposing rotation of the molecules is small
even though the forces opposing translational motion may be very
large. The particular equation for the calculation of the time of
relaxation given above obviously does not apply to solids.

In discussing the three types of polarizations which have been
considered thus far, it has been pointed out that the magnitude of
the dielectric constant depends upon the polarizability of the material.
Each type of polarization makes a contribution to the dielectric
constant if the measuring frequency is considerably below its relax-
ation-frequency. However, if the frequency of the applied field used
for measuring the dielectric constant is too high the presence of
polarizations with low relaxation-frequencies will not be detected.
Thus the refractive index of water in the visible spectrum is 1.3 and
therefore gives no evidence whatever of the presence of permanent
dipoles. This is due to the fact that the H2O molecules do not change
their orientations rapidly enough to allow fields which alternate in
direction as rapidly as those of light to cause an appreciable deviation
from the original random orientation which prevails in the absence of
an applied field.

The band of frequencies in which the dielectric constant decreases
with increasing frequency because of inability of the polarization to
form completely in the time available during a cycle, is called a region
of absorption or of anomalous dispersion. The discussion of this
characteristic of dielectric materials forms an important part of
dielectric theory. The term anomalous dispersion is no doubt usually
thought of in connection with the anomalous dispersion of light : when
the refractive index of light decreases with increasing frequency the

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 507

material is said to display anomalous dispersion in the range of fre-
quencies concerned. However, in a paper published in 1898 Drude 7
applied this term to the decrease of dielectric constant with increasing
frequency in the electrical range of frequencies. The justification
for this extension of the original application of the term is very direct
for electromagnetic theory shows that the dielectric constant and the
refractive index of a material are connected by the general relationship
e = n 2 whatever the frequency of the electromagnetic disturbance.
As the dispersion of light by a prism is due to the variation of its re-
fractive index with frequency, the use of the expression anomalous
dispersion to refer to the decrease of dielectric constant with increasing
frequency is consistent and has become generally accepted.

Interfacial Polarizations

The polarizations thus far considered are the main types to be
expected in a homogeneous material. They depend upon the effect
of the applied field in slightly displacing electrons in atoms, in slightly
distorting the atomic arrangement in molecules and in causing a slight
deviation from randomness in the orientation of polar molecules.
The remaining types of polarization are those resulting from the
heterogeneous nature of the material and are called interfacial polariza-
tions. Interfacial polarizations must exist in any dielectric made up
of two or more components having different dielectric constants and
conductivities except for the particular case where 6172 = €271, 7 being
the conductivity 8 and the subscripts referring to the two components.
Heterogeneity in a dielectric may be due to a number of causes, and
in the case of practical insulating materials is probably the rule rather
than the exception. Impregnated paper condensers and laminated
plastics are obvious examples of heterogeneous dielectrics. Paper
is itself a heterogeneous dielectric, consisting of water and cellulose.
In all probability the plastic resins are also heterogeneous, and cer-
tainly so if they contain fillers. Ceramics, being mixtures of crystalline
and glassy phases, are also heterogeneous.

The simplest case of interfacial polarization is that of the two-layer
dielectric, that is, a composite dielectric made up of two layers, the
dielectric constants and conductivities of which are different. Max-
well showed that in such a system the capacity was dependent upon
the charging time. This is due to the accumulation of charge at the
interface between the two layers, for this charge must flow through a

7 P. Drude, Ann. d. Physik, 64, 131 (1898), " Zur Theorie der anomalien elektrischen
Dispersion."

8 In this expression 7 represents the total a.c. conductivity, a quantity which
depends on the frequency.

508 BELL SYSTEM TECHNICAL JOURNAL

layer of dielectric whose resistance may be high enough that the inter-
face does not become completely charged during the time allowed for
charging. For the alternating current case this implies a decrease of.
capacity with increasing frequency, which is equivalent to the anoma-
lous dispersion which has been described for the case of dipole polariza-
tions. It should be particularly emphasized that the term anomalous
dispersion describes a type of variation of dielectric constant with
frequency which can be produced by a number of different physical
mechanisms.

The two-layer dielectric is of less interest than a generalization of
this type of polarization which includes heterogeneous systems com-
posed of particles of one dielectric dispersed in another. This type
of heterogeneous dielectric is of considerable importance since such
systems represent the actual structure of many practical dielectrics.
Such a generalization of the two-layer dielectric has been made by
K. W. Wagner 9 who developed the theory for the case of spheres of
relatively high conductivity dispersed in a continuous medium of low
conductivity. The conditions for the existence of an interfacial
polarization are, as in the two-layer case, that eftx + «27i» where the
symbols have the significance just given. This type of polarization,
which is variously referred to as an interfacial polarization, an ionic
polarization and a Maxwell-Wagner polarization, shows anomalous
dispersion like other absorptive polarizations. When the particle
size is small as compared with the electrode separation it may be
treated as a uniformly distributed polarization.

The magnitude and time of relaxation of interfacial polarizations
are determined by the differences in e and 7 of the two components.
There is a widely prevalent opinion that this type of polarization
always has such long relaxation-times as to be observed only at very
low frequencies. While this is true for mixtures of very low-con-
ductivity components, the general equations show that for the case
where one component has a high conductivity — for example equal to
that of a salt solution — the dispersion may occur in the radio frequency
range.

Several special types of interfacial polarization have been proposed
to explain the dielectric properties of various non-homogeneous di-
electrics where something regarding the nature of the inhomogeneity
is known. The dielectric constant of cellulose, for example, receives
a contribution from an interfacial polarization due to the water and
dissolved salt which it contains. Experimental evidence indicates
that an aqueous solution of various salts is distributed through the

» K. W. Wagner, Arch.f. Elektrotechn., 2, pp. 374 and 383.

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 509

cellulose in such a way as to form a reticulated pattern which may
correspond to the pattern formed by the micelles or to some feature
of it. An interesting feature of this structure is that the conductance
of the aqueous constituent can be changed by varying the moisture
content or the salt content of the material and the effect on the di-
electric constant observed. 10

Frequency Dependence of Dielectric Constant
As has been pointed out, each of the different types of polarization
may contribute to the dielectric constant an amount depending upon
the polarizability and its time of relaxation. The upper curve in Fig.
1 shows schematically the variation of the dielectric constant (or of the
square of the refractive index) for a hypothetical material possessing
an interfacial polarization with relaxation-frequency in the power
range, a dipole polarization with relaxation frequency in the high
radio frequency range and atomic and electronic polarizations with
dispersion regions in the infra-red and visible respectively. If polariza-
bility were plotted, instead of e (or w 2 ), the curves would be of the same
general form but of different magnitudes, because of a relationship
between the two given earlier.

At the low-frequency side of Fig. 1, the dielectric constant curve
has its highest value, usually called the static or zero-frequency
dielectric constant. Here all of the polarizations have time to form
and to contribute their full amount to the dielectric constant. With
increasing frequency e begins to decrease as the relaxation-frequency
of the interfacial polarization is approached and reaches a constant
lower value (called the infinite-frequency dielectric constant) when
the applied frequency is sufficiently above the relaxation-frequency of
the polarization that it has not time to form appreciably. It is this
decrease of e with frequency which is called anomalous dispersion. The
horizontal arrows across the top of Fig. 1 indicate the frequency region
in which the various types of polarizations indicated are able to form
and contribute to the dielectric constant.

At still higher frequencies we see that e again decreases as the
relaxation-frequency of the dipole polarization is approached, and
again reaches a constant lower value as the frequency becomes too
high for the field to affect appreciably the orientation of dipoles.
This second region of anomalous dispersion is similar to the first,
which was due to interfacial polarizations. It has been shown as
occurring at a higher frequency, but it should be emphasized that the
frequency ranges chosen to illustrate anomalous dispersion in Fig. 1
i° Murphy and Lowry, Jour. Phys. Chem., 34, 594 (1930).

510 BELL SYSTEM TECHNICAL JOURNAL

are purely arbitrary. Anomalous dispersion due to dipole polariza-
tions has been observed at power frequencies while that due to inter-
facial polarizations has been observed at radio frequencies. The two
types of polarizations may in fact give rise to anomalous dispersion
in the same frequency range in a given dielectric.

Proceeding to still higher frequencies in Fig. 1 other regions of
dispersion appear in the infra-red and visible spectrum. These
regions show a combination of normal optical dispersion, in which the
dielectric constant, or better now the refractive index, increases with
frequency, and anomalous dispersion in which it decreases. The
dispersion in the visible range of frequencies is predominantly normal
(anomalous dispersion being confined to relatively narrow frequency
bands) whereas in the electrical range the reverse is true, normal
dispersion not being observed ; the infra-red represents an intermediate
region. Dipole and interfacial polarizations are not represented in
the dispersion in the optical range, the dielectric constant (or refractive
index) in the visible being due to electronic polarizations and in the
infra-red to electronic and atomic polarizations.

The curves plotted in Fig. 1 are merely schematic and the relative
magnitudes of the different contributions to the dielectric constant
are therefore arbitrary. However, experimental results indicate that
the contribution €e of the electronic polarization to the dielectric
constant is limited to values between 2 and 4 except for certain in-
organic materials, since very few organic solids or liquids are known
which have refractive indices in the visible spectrum which are greater
than 2 or less than 1.4. The contribution 6a of atomic polarizations
to the dielectric constant is in general small and is usually negligible,
as has been indicated on the curve, although the possibility exists of
special cases occurring in which the infra-red refractive indices are
very high. The contributions e P and e/ of dipole and interfacial
polarizations to the dielectric constant may vary greatly from one
material to another, depending upon the symmetry of the molecule
and the physical structure of the dielectric. From the above men-
tioned limitations on the contribution to the dielectric constant which
can be expected from electronic and atomic polarizations, it is apparent
that the explanation of values of e higher than 3 to 4, at least in organic
materials, requires the existence of some absorptive polarization such
as arises from dipoles or interfacial effects. Thus all of the liquids
which have high dielectric constants such as H2O (78), alcohol (24),
nitrobenzene (34) have been shown to contain polar molecules.

The lower part of Fig. 1 shows a maximum in the absorption for
each type of dielectric polarization. The absorption, at least in the

DIELECTRIC PROPERTIES OF INSULATING MATERIALS 511

electrical frequency range, is due to the dissipation of the energy of the
field as heat because of the friction experienced by the bound charges
or dipoles in their motion in the applied field in forming the polariza-
tions. The theory of dispersion shows that the dielectric constant and
absorption are not independent quantities but that the absorption
curve can be calculated from the dielectric constant vs. frequency
curve and vice versa. The absorption maximum is greatest for those
materials showing the greatest change in dielectric constant in passing
through the dispersion region. Thus a material having a high di-
electric constant must have a large dielectric loss at the frequency at
which c has a value half way between its low and high-frequency values.
Though the quantum theory is necessary for the explanation of many
optical and electrical phenomena a simple explanation, sufficient for
our purposes, of the general form of the curves of dielectric constant vs.
frequency in the infra-red and visible spectrum may be given in terms
of the Lorentz theory of optical dispersion. In this theory the form
of the dispersion curves depends upon the variation with frequency of
the relative importance of the inertia of the typical electron and of the
frictional forces and restoring forces acting upon it. For electronic
polarizations the frictional or dissipative force is negligible, except
in the narrow frequency interval included in the absorption band, and
the inertia and restoring force terms predominate. For the atomic
polarizations the frictional force is larger and the absorption region
extends over a wider interval of frequencies. For dipole and inter-
facial polarizations the influence of inertia is entirely negligible as
compared with the frictional or dissipative forces so that in effect these
polarizations may be thought of as aperiodically damped.

Temperature Dependence of Dielectric Constant
The dielectric constant of a material is a constant only in the ex-
ceptional case. Besides the variation with frequency which has been
considered the dielectric constant varies with temperature. Elec-
tronic polarizations may be considered to be unaffected by the tempera-
ture. The refractive index does indeed change with temperature
but this is completely accounted for by the change of density, and the
molar refraction is independent of temperature. The atomic and
ionic vibrations are, however, affected by temperature, the binding
force between ions or atoms being weakened by increased temperature.
This factor of itself would yield a positive temperature coefficient for
the atomic polarizations but the decrease in density with the increase
in temperature acts in the opposite direction. The result is that
calculation of the temperature coefficient of atomic polarizations

'

512 BELL SYSTEM TECHNICAL JOURNAL

usually yields zero or slightly positive values. What experimental
data there are indicate small positive temperature coefficients for
atomic polarizations.

One of the principal achievements of the Debye theory of dipole
polarizations has been the manner in which it explains the large
negative temperature coefficients of polarization of many liquids.
Debye showed that the variation of polarization with temperature
could be expressed by the relation P = A + (B/T), in which the
constant A is a measure of the instantaneous polarizations which are
independent of temperature and B is a measure of the dipole polariza-
tions. In a liquid or gas the molecules are continuously undergoing
both translational and rotational motion, and the result of this thermal
motion is to maintain a random orientation of molecules. The action
of the electric field in aligning the dipoles is opposed by the thermal
motion which acts as an influence tending to keep them oriented at
random. As the temperature decreases, the thermal energy becomes
' smaller and the dipole polarization becomes larger, resulting in a

negative temperature coefficient of dielectric constant.

The effect of temperature upon interfacial polarizations has not
been experimentally investigated to an extent at all comparable with
that of dipole polarizations. However, interest in the interfacial or
ionic type of polarization has increased considerably in the past few
years, and it has applications of some importance. Among these is
diathermy which is becoming of considerable importance as a thera-
peutic agency.

The foregoing qualitative description of the behavior of the di-
electric constant and the type of information regarding molecular
structure which has been derived from it will be followed in the next
section by the derivation of some of the quantitative relationships
which are common to all polarizations of the absorptive type.